Why would Arctic melt cause sea level to rise?

I have heard people say that the melting of the Arctic ice cap would raise the sea level a lot, but in a glass of ice cubes and water, the ice cubes' melting does not cause the water to overflow the glass, so why is this being presented as a consequence of climate change?

Climate change has caused the polar ice caps to reduce in size, but as you rightly say, this does not cause a rise in sea levels. However this can have an effect on the overall temperature and currents of sea water. The reduced size of the artic ice cap means the gulf stream can travel further north before it cools. This results in the north Atlantic ocean being warmer, thus fish such as the Cod move north to follow the lower temperature and, of course, their food.

A reduced Artic Ice cap also allows the sun to warm the waters because there is less ice to reflect the sun's rays.

If the whole North Atlantic region becomes warmer, then there is a chance that the glaciers may melt quicker. Should the glaciers of Greenland melt, then, as they are on land, sea levels will rise.

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If brains were made of dynamite, I wouldn't have enough to blow my nose.

When water heats up, it expands. Thus, the most readily apparent consequence of higher sea temperatures is a rapid rise in sea level. Sea level rise causes inundation of coastal habitats for humans as well as plants and animals, shoreline erosion, and more powerful storm surges that can devastate low-lying areas.

It's important to distinguish the Arctic and Antarctic here.The ice at the North pole is floating. But the ice at the South pole is on land, if it melts it runs down into the sea and raises the level.The same is also true for a lot of ice in places like Greenland.

Thanks. I know the Antarctic is land based ice but i was just intrigued by those who cite the Arctic as the cause of this rise in sea level. And if the warming of the water is one effect, wouldn't this result in more evaporation that would reduce the rise in sea level? perhaps more rain on land?

Evaporation is part of a cycle of evaporation and rain. So, perhaps more clouds would mean less water in the sea, but I would assume that would be a minimal effect. However, humidity is normally reported as a relative humidity. As the temperature rises, the amount of water carrying capacity of the air increases. This could potentially be significant, except that we're only talking about a couple of degrees change.

As RD and BC mentioned, we're experiencing sea level rise due to expansion of the water, and land based glacier melt.

Some estimates indicate that the ocean basins are also slowly expanding due to less weight of glaciers on the land since the early Holocene. Many of the agencies who are reporting the sea levels also add a factor into their calculations based on this estimate of an expanding ocean basin, and are reporting sea levels higher than actually observed.